To determine which graph represents the function [tex]\( f(x) = \frac{x^2 - 49}{x + 7} \)[/tex], let's go through a detailed, step-by-step analysis and simplification of the function.
1. Identify the numerator and denominator:
- The numerator is [tex]\( x^2 - 49 \)[/tex].
- The denominator is [tex]\( x + 7 \)[/tex].
2. Factor the numerator:
- Notice that [tex]\( x^2 - 49 \)[/tex] is a difference of squares.
- So, [tex]\( x^2 - 49 = (x - 7)(x + 7) \)[/tex].
3. Rewrite the function:
- Substitute the factored form of the numerator back into the function:
[tex]\[
f(x) = \frac{(x - 7)(x + 7)}{x + 7}.
\][/tex]
4. Simplify the function:
- Assuming [tex]\( x \neq -7 \)[/tex] (to avoid division by zero), the common factor [tex]\( x + 7 \)[/tex] in the numerator and denominator cancels out:
[tex]\[
f(x) = x - 7.
\][/tex]
Thus, the simplified function is [tex]\( f(x) = x - 7 \)[/tex], except at [tex]\( x = -7 \)[/tex], where the original function [tex]\( f(x) \)[/tex] is undefined.
5. Analyze the simplified function:
- The function [tex]\( f(x) = x - 7 \)[/tex] is a linear function with a slope of 1 and a y-intercept of -7.
- The graph will be a straight line passing through the point (0, -7).
- At [tex]\( x = -7 \)[/tex], there will be a hole in the graph, because the original function is undefined at this point.
6. Graph of the function:
- It will be a straight line starting from the y-intercept of -7, moving with a slope of 1.
- There will be a hole at the point [tex]\((-7, -14)\)[/tex], since [tex]\( f(-7) \)[/tex] is undefined.
In summary, the function [tex]\( f(x) = \frac{x^2 - 49}{x + 7} \)[/tex] simplifies to [tex]\( f(x) = x - 7 \)[/tex], with a hole at [tex]\( x = -7 \)[/tex]. The graph you are looking for should show a straight line with a slope of 1 and a y-intercept at -7, including a hole at the point where [tex]\( x = -7 \)[/tex].
